The shapes of many leaves and flowers are determined, at least in part, by their
inhomogeneous growth. Additional growth at the edges of leaves, for example,
result in saddle-splay curvature and, ultimately, wrinkling. Recent experiments on
thin, polymer films imprinted with a predefined pattern of inhomogeneous swelling
provide a controlled, experimental playground for describing how swelling-induced
buckling leads to a prescribed three-dimensional shape. For example, one expects
a sufficiently thin sheet to buckle into a shape that eliminates most of its in-plane
strain. Though this is always possible locally, a particular swelling pattern may be
either globally frustrated, having no stress-free shapes even for vanishing
thickness, or lead to a large degeneracy of stress-free shapes. In both cases, the
bending energy remains important even for very thin sheets.
I will describe theoretical work on disks and narrow ribbons with swelling-induced,
negative Gaussian curvature. Perhaps surprisingly, when the prescribed Gaussian
curvature is constant, there are families of stress-free and nearly stress-free
shapes, none of which seem to appear in experiments. To understand this behavior,
I will identify regimes in which the minimal energy ribbon shape can be determined
and discuss the role of stretching and bending energies. We will consider both strips
and closed ribbons. [Preview Abstract]

An elastic membrane that is forced to reside in a container of
a slightly smaller size will deform and, upon further volume
reduction, will eventually crumple. Previous studies have
focused on the onset of the crumpled state by analyzing the
mechanical response and stability of a developable conical
surface (d-cones) that can be described by a single-valued
function, while others have simulated the highly packed regime,
neglecting the importance of connectivity of the membrane. Here
we present a study in which experiments, numerical simulation
and analytic work are used to show that the emergence of new
regions of high stretching is a generic outcome when a self-
avoiding membrane is subject to a severe geometrical
constraint. Consequently, an anomalous mechanical response,
characterized by a series of peaks in the force-deformation
curve, appears as the membrane is squeezed. Our findings
emphasize the role of self-avoidance, connectivity and friction
as the key factors defining the morphology and response of a d-
cone from its formation to its final fate.
[Preview Abstract]

We present two experiments in which the interplay between
stretching and bending modes in thin elastic plates plays an
important role. The first experiment is motivated by the
understanding of plant leaf shape (e.g. Maize leaves, grass).
During growth, many leaves unfold to become flat while their
bottom is still attached to the cylindrical stem. We investigate
the mechanical analogue of this unfold length and address the
question: what is the persistence length of a curvature applied
at one end of a flat elastic strip? Simple scaling arguments are
compared to experiments and numerical simulations using Surface
Evolver. The second experiment concerns the snap-buckling
instability of highly deformed plates, in relation to the noise
of crumpling. Using high-speed video and
three-dimensional shape reconstruction, we show that
snap-buckling instabilities in thin plates are non-homogeneous
and occur via the very fast propagation of an elastic defect. The
speed of the transition and the acoustic signature of the snap
are mainly controlled by the defect size. [Preview Abstract]

We study experimentally the shapes that result when a circular
sheet of polystyrene film (diameter~3mm, thickness t=75-500
nm) is placed on the surface of an initially spherical water
droplet. The competition between surface energies of the fluid
interfaces and the elastic stresses in the sheet leads to
interesting compromises between their energetically preferred
shapes. We report the progression of features that result from
continuously varying the droplet volume (and curvature) for
various film thicknesses and sizes. For small droplet curvature,
the sheet is smoothly stretched. It then develops smooth radial
wrinkles at the edges. At larger drop volumes, the azimuthal
symmetry is further broken as some wrinkles turn into focused
d-cone-like points, and the droplet develops a faceted polygonal
flat ``table-top.'' [Preview Abstract]

We use X-ray CT scanning to resolve in 3-dimensions the
conformation of aluminum sheets with thickness t=25 microns
crumpled into spherical balls with average volume fractions,
$\phi $ ranging from 0.06 to 0.22. We have previously reported an
inhomogeneous distribution of mass in the volume: the volume
fraction increases with radius so that the sphere is densest at
its surface. We now report on the geometry of the sheet, in
particular we report on the distribution of Gaussian and mean
curvature in the sample, with a view to quantifying the
arrangement of regions of stress-focusing. A new feature
apparent in the images is an unusually strong degree of stacking
into multilayered facets. We quantify layering in the sample by
reporting on the local nematic ordering of the sheet normals. [Preview Abstract]

We present a theoretical study of free non-Euclidean plates with
a disc
geometry and a prescribed metric that corresponds to a constant
negative
Gaussian curvature. We take the equilibrium configuration taken
by the these
sheets to be a minimum of a F\"{o}ppel Von-K\`{a}rm\`{a}n type
functional in
which configurations free of any in plane stretching correspond
to isometric
embeddings of the metric. We show for all radii there exists low
bending
energy configurations free of any in plane stretching that obtain
a periodic
profile. The number of periods in these configurations is set by the
condition that the principle curvatures of the surface remain
finite and
grows approximately exponentially with the radius of the disc. [Preview Abstract]

The self-assembly of patterns from isotropic initial states is a
major driver of modern soft-matter research. This avenue of
study is directed by the desire to understand the complex physics
of the varied structures found in Nature, and by technological
interest in functional materials that may be derived through
biomimicry. In this work we show how a simple striped phase can
respond with significant complexity to an appropriately chosen
perturbation. In particular, we show how a buckled elastic plate
transitions into a state of stress localization using a simple,
self-assembled variation in surface topography. The collection
of topographic boundaries act in concert to change the state from
isotropic sinusoidal wrinkles, to sharp folds or creases
separated by relatively flat regions. By varying the size of the
imposed topographic pattern or the wavelength of the wrinkles, we
construct a state diagram of the system. The localized state has
implications for both biological systems, and for the control of
non-linear pattern formation. [Preview Abstract]

Confining elastic sheets often results in the formation of singular,
stress-focusing structures: ridges and vertices in which strain is
localized, allowing the sheet to reach a developable shape in the limit of
vanishing thickness. The formation of such developable shapes through a
network of singularities may become impossible, however, when certain types
of geometric constraints are imposed on the sheet. One may ask: Are there
other fundamental types of stress distribution that govern patterns on
elastic sheets under such conditions? In particular -- what is the nature of
transition zones between singular and smoothly bent structures that may
emerge in separate parts of a stressed sheet?
We will address these questions through simple model systems that
demonstrate the emergence of nontrivial shapes under such conditions. [Preview Abstract]

Hyperbolic elastic Non-Euclidean plates are plates whose two
dimensional
target metric prescribes a negative Gaussian curvature. The
equilibrium
configurations of such, axi-symmetric, bodies are known to
consist of
multi-scale wave cascades.
Using environmentally responsive gels, we experimentally study
the change of
the wavy patters with the thickness of the discs and with their
radius. We
provide the scaling of the number of nodes with respect to these
parameters
and show that, as the disc thickness decreases, the bending
content of the
discs sharply increases. This increase is compensated by a
reduction of
stretching content, due to the refinement of scales. [Preview Abstract]

We consider an elastic shell with two coexisting components
having different bending rigidities and elastic constants. We
explore the low-energy configuration of the shell when the
relative fraction of the two components and their elastic
constants vary. We analyze different domain patterns associated
with the shape of the shell. We also study the effect of a line
tension term between the two components. We show how the
relation between the morphology of the shell and the domain
patterns lead to a rich variety of structures. In particular we
discuss the role of the buckling instability in this model with
heterogeneous components. [Preview Abstract]

Vertical compression of an elastic thin-walled box is explored.
Such a compression displays three successive regimes: linear,
buckled and collapsed. Analogy of the buckled regime to thin-film
blisters is demonstrated. The compression force is shown to reach
its maximum at the end of that regime, after which the box
collapses displaying features (e.g. ridges) typical of crumpling
of thin sheets. These qualitative findings are confirmed by
numerical simulations based on a discrete element method, and
implications are drawn on the box compression strength. [Preview Abstract]

This work focuses on the wrinkle-to-fold transition in glassy and
elastomeric thin films. In biological systems, the process of
folding is critical to morphogenesis, defining such features as
the neural folds in embryonic development. In this paper, we
examine the deformation of an axisymmetric sheet and quantify the
force required to generate a fold. A thin film draping over a
point of contact will wrinkle due to the strain imposed by the
point and the underlying substrate. The wrinkle wavelength is
dictated by a balance of material properties and geometry, and
scales with film thickness to the three-fourths power. At a
critical strain the stress in the film will localize, causing
hundreds of wrinkles to collapse into several discrete folds. We
measure the energy of formation for a single fold and observe
that it scales linearly with film thickness. We predict that the
onset of folding, from a critical force or displacement, scales
as the thickness to the one-fifth power. The folds act as
disclinations in the film causing the stress in the film to
increase, thereby decreasing the wavelength of the remaining
wrinkles. The number of folds that form from wrinkle collapse
appears to be constant over several orders of magnitude in film
thickness and elastic modulus.
[Preview Abstract]